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Symmetry classification using noncommutative invariant differential operators. (English) Zbl 1107.35011

This paper is devoted to the problem of point symmetry classification of differential equations. Authors show how to perform this classification for a class of PDEs in a \({\mathcal G}\)-invariant way, where \({\mathcal G}\) is a known Lie pseudogroup of point transformations preserving the class of PDEs. Their approach is based on some differential elimination procedure that completes the system of PDEs defining classical symmetries by adjoining compatibility conditions. Each step of the procedure is invariant w.r.t. \({\mathcal G}\). The approach is applied to a class of nonlinear diffusion convection equations \(v_x=u\), \(v_t=B(u)u_x-K(u)\), which is invariant w.r.t. some equivalence pseudogroup \({\mathcal G}\). In this example the complexity of the calculations is much reduced by the use of a \({\mathcal G}\)-invariant differential operator.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35N10 Overdetermined systems of PDEs with variable coefficients
58J70 Invariance and symmetry properties for PDEs on manifolds
35K55 Nonlinear parabolic equations

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